How this half-life calculator works
Half-life is the time required for a quantity to fall to half of its starting value. It is central to radioactive decay, but the same exponential-decay math also describes drug elimination, capacitor discharge and many other processes. Given an initial quantity N₀ and a half-life T½, the amount remaining after an elapsed time t is:
N = N₀ × (1/2)(t / T½)
The exponent t / T½ is simply the number of half-lives that have passed. After one half-life half remains, after two a quarter remains, and so on. The calculator also reports two related constants. The decay constant λ = ln(2) / T½ gives the fractional rate of decay per unit time, so the same decay can be written N = N₀ × e−λt. The mean lifetime τ = 1 / λ = T½ / ln(2) ≈ 1.4427 × T½ is the average time a single particle survives before decaying.
Switch the calculator to solve for elapsed time and it inverts the formula: t = T½ × log₂(N₀ / N). Keep the half-life and elapsed time in the same time unit — the dropdown applies one unit to both so the ratio t / T½ stays dimensionless.
Remaining percentage after each half-life
| Half-lives elapsed | Fraction remaining | Percent remaining |
|---|---|---|
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
Reference note: decay is exponential, so the quantity approaches zero but never reaches it exactly. This is a general educational physics and chemistry tool and is not medical, safety or professional advice.
Frequently asked questions
- What is half-life?
- Half-life (T½) is the time it takes for a decaying quantity to fall to half its starting value. After one half-life half remains, after two a quarter remains, and so on. It is a property of the decay process and does not depend on how much you started with.
- How do I calculate the remaining amount?
- Use N = N₀ × (1/2)^(t/T½), where N₀ is the initial quantity, t is the elapsed time and T½ is the half-life. The exponent t/T½ is the number of half-lives elapsed. Keep t and T½ in the same time unit so the ratio is correct.
- What is the decay constant?
- The decay constant λ is the probability per unit time that a given particle decays. It is related to the half-life by λ = ln(2)/T½ ≈ 0.6931/T½. A larger decay constant means faster decay and a shorter half-life.
- How many half-lives until the amount is nearly gone?
- Each half-life removes half of what remains, so the amount approaches zero but never reaches it exactly. After about 7 half-lives less than 1% remains, and after about 10 half-lives roughly 0.1% remains, which is often treated as negligible.
- What is mean lifetime?
- The mean lifetime τ is the average time a particle survives before decaying. It equals 1/λ, or T½/ln(2) ≈ 1.4427 × T½. The mean lifetime is always longer than the half-life because a few long-lived particles raise the average.
- Does half-life apply to medications too?
- Yes. In pharmacology the biological or elimination half-life describes how long it takes the body to reduce a drug's concentration by half, and the same N = N₀ × (1/2)^(t/T½) math applies. This page is a general educational tool and not medical advice.